Fuchs Michael, Steel Mike, Zhang Qiang
Department of Mathematical Sciences, National Chengchi University, Taipei, 116, Taiwan.
Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand.
Bull Math Biol. 2025 May 7;87(6):69. doi: 10.1007/s11538-025-01444-y.
Phylogenetic networks provide a more general description of evolutionary relationships than rooted phylogenetic trees. One way to produce a phylogenetic network is to randomly place k arcs between the edges of a rooted binary phylogenetic tree with n leaves. The resulting directed graph may fail to be a phylogenetic network, and even when it is it may fail to be a tree-child or normal network. In this paper, we first show that if k is fixed, the proportion of arc placements that result in a normal network tends to 1 as n grows. From this result, the asymptotic enumeration of normal networks becomes straightforward and provides a transparent meaning to the combinatorial terms that arise. Moreover, the approach extends to allow k to grow with n (at the rate ), which was not handled in earlier work. We also investigate a subclass of normal networks of particular relevance in biology (hybridization networks) and establish that the same asymptotic results apply.
系统发育网络比有根系统发育树能更全面地描述进化关系。生成系统发育网络的一种方法是在具有(n)个叶子节点的有根二叉系统发育树的边之间随机放置(k)条弧。得到的有向图可能不是系统发育网络，即使它是系统发育网络，也可能不是树孩子网络或正规网络。在本文中，我们首先表明，如果(k)是固定的，随着(n)的增长，导致正规网络的弧放置比例趋于(1)。基于这个结果，正规网络的渐近枚举变得直接明了，并为出现的组合项赋予了清晰的含义。此外，该方法可以扩展到允许(k)随(n)增长（增长率为 ），这在早期工作中并未涉及。我们还研究了生物学中特别相关的正规网络子类（杂交网络），并确定相同的渐近结果适用。
